Polynomial Calculator: Add, Multiply, Divide, Factor and Find Roots With Steps

Type a polynomial and say what you want: add, multiply, divide, factor or find the roots. The solver names the method, works through it line by line and checks the answer by multiplying back.

Solve Any Polynomial Problem With the Mathvelo Solver

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Pick a polynomial taskWrite powers with ^, like x^3. Say what you want first: divide, factor, expand or find the roots.
Dividedivide x^3-6x^2+11x-6 by x-1Factorfactor 2x^3-3x^2-11x+6Find rootsfind the roots of x^3-6x^2+11x-6Multiplyexpand (x+2)(x^2-3x+4)Add(3x^2+2x-1)+(x^2-5x+4)Remainderremainder of x^3-2x+5 divided by x-2
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What a Polynomial Is and How to Read One

A polynomial is a sum of terms like 4x³ or −7x, each a number times a whole-number power of x. The highest power is the degree, the number in front of it is the leading coefficient, and writing the terms from highest power to lowest is called standard form. In 2x³ − 3x² − 11x + 6 the degree is 3, the leading coefficient is 2 and the constant term is 6.

The degree tells you how many roots to expect. A degree-3 polynomial has exactly three roots counting repeats and complex ones, and at most two turning points on its graph.

Polynomial Long Division and Synthetic Division

Long division works like the one from primary school: divide the leading terms, multiply back, subtract, bring down. When the divisor is x − r, synthetic division does the same job using only the coefficients, which is much faster by hand.

Divide x³ − 6x² + 11x − 6 by x − 1Try it in the solver ↑

(x³ − 6x² + 11x − 6) ÷ (x − 1)

  1. Set up synthetic division with r = 1 and coefficients 1, −6, 11, −6.
  2. Bring down the 1.
  3. Multiply by 1 and add: −6 + 1 = −5. Then 11 + (−5) = 6. Then −6 + 6 = 0.
  4. The bottom row 1, −5, 6 gives x² − 5x + 6, and the remainder 0 means x − 1 is a factor.
  5. Factor the quotient: x² − 5x + 6 = (x − 2)(x − 3).

Answer: x² − 5x + 6, so the roots are 1, 2 and 3

The Remainder Theorem is the shortcut hiding in that table: dividing by x − r leaves the remainder p(r). If it is zero, r is a root.

Finding Roots With the Rational Root Theorem

For a polynomial with whole-number coefficients, every rational root has the form p/q where p divides the constant term and q divides the leading coefficient. That turns an open-ended search into a short list.

Factor 2x³ − 3x² − 11x + 6Try it in the solver ↑

factor 2x³ − 3x² − 11x + 6

  1. The constant is 6 and the leading coefficient is 2, so candidates are ±1, ±2, ±3, ±6, ±1/2 and ±3/2.
  2. Test x = 3: 2(27) − 3(9) − 11(3) + 6 = 54 − 27 − 33 + 6 = 0, so x = 3 is a root.
  3. Divide by (x − 3) to get 2x² + 3x − 2.
  4. Factor the quadratic: 2x² + 3x − 2 = (2x − 1)(x + 2).
  5. Put it together: (x − 3)(2x − 1)(x + 2). The roots are 3, 1/2 and −2.

Answer: (x − 3)(2x − 1)(x + 2)

Adding and Multiplying Polynomials

Adding is just combining like terms. Multiplying means every term in the first factor meets every term in the second, then like terms are combined.

Multiply (x + 2)(x² − 3x + 4)Try it in the solver ↑

(x + 2)(x² − 3x + 4)

  1. Distribute x: x³ − 3x² + 4x.
  2. Distribute 2: 2x² − 6x + 8.
  3. Add the two lines.
  4. Combine like terms: x³ + (−3x² + 2x²) + (4x − 6x) + 8.

Answer: x³ − x² − 2x + 8

End Behaviour, Multiplicity and the Graph

Leading termLeft endRight end
Even degree, positive coefficient (x²)upup
Even degree, negative coefficient (−x⁴)downdown
Odd degree, positive coefficient (x³)downup
Odd degree, negative coefficient (−x³)updown

Multiplicity shapes the crossing. A root that appears once makes the graph cross the axis. A root that appears twice, as in (x − 2)², makes it bounce off. A triple root flattens and crosses, like x³ at the origin.

Original Data: How Often Does a Cubic Have a Rational Root?

Counting all 123,462 cubics ax³ + bx² + cx + d with integer coefficients from −9 to 9 and a ≠ 0, using the rational root theorem (candidates p ÷ q with p dividing d and q dividing a). Only 15.2% have a rational root. For the other 85%, factoring by hand fails and a numeric or formula method is the only way in. That is why “try factoring first” is good advice for textbook problems and poor advice for real data.

Our Take: What Is Overrated, Underrated and Plain Wrong

Overrated: hunting for roots by trial. The rational root theorem narrows the candidates, but a calculator or a graph is faster for anything past a cubic.

Underrated: synthetic division. Once you know one root it reduces the degree in a few lines and is much less error-prone than long division.

Plain wrong: the belief that every polynomial has a root formula. Abel and Ruffini showed in the 1800s that general quintics (degree 5) cannot be solved with radicals.

Polynomial Mistakes That Cost Marks

  • Leaving a gap in synthetic division. A missing power needs a 0 coefficient, so x³ − 1 starts 1, 0, 0, −1.
  • Dividing by x + 3 but using r = 3. The root is r = −3.
  • Stopping once one factor is found. Keep factoring until every piece is linear or irreducible.
  • Forgetting to check by multiplying back out.

Why Students Get Stuck on Polynomials, and What a Good Calculator Should Do

Searching for help with polynomials usually starts late at night with one problem that will not work out. Long expressions are easy to mis-copy, and roots are hard to guess. A good calculator fixes that by showing each step, checking its own answer and explaining the idea in plain words. This section covers what students struggle with, what they want, and how this page responds.

In short: students need to add, multiply, divide and find roots without losing terms, not just see a final number, and a calculator earns trust only when its steps can be checked.

Understand the frustrations and desires

What they feel, and what they want

Each frustration points to a wish: a clearer step, a built-in check, an explanation in plain words.

  • Wants like terms collected in a clear line
  • Wants the division steps shown
  • Wants roots and factors together

Build the page around their intent

Each intent has a place on this page

The page follows intent, not features, so each need has a clear starting point.

  • I need an answer now: type the problem into the calculator at the top of this page.Multiply (x + 2)(x^2 - 3x + 4)
  • I need to understand: the sections below explain the idea in plain words and list the common mistakes.
  • I need to check my work: every step is shown, so the first line that differs from yours is the one to study.
  • I need practice: open the solver, tap + and choose Create Practice Test.
  • I need it explained aloud: in the solver, tap + and choose Create Video.

Polynomial: Questions People Ask

What is a polynomial?

An expression made of terms with whole-number powers of a variable, such as 3x² − 5x + 2.

How do I find the roots of a polynomial?

Look for factors with the Rational Root Theorem, divide them out, and solve what is left. For a quadratic remainder use the quadratic formula.

What is the difference between long and synthetic division?

Synthetic division is a shortcut that works only when dividing by x − r. Long division works for any polynomial divisor.

How many roots can a polynomial have?

A degree-n polynomial has exactly n roots counting multiplicity and complex roots, and at most n real roots.

What does the remainder theorem say?

Dividing p(x) by x − r leaves remainder p(r). A remainder of zero means r is a root.

Can the solver factor polynomials with no nice roots?

It will find the factors it can over the rationals and tell you when the rest is irreducible. For numeric roots it gives decimals.

How do I divide polynomials?

Use long division or, for a divisor x − r, synthetic division. Divide the leading terms, multiply back, subtract and repeat until the remainder has lower degree.

What is the degree of a polynomial?

The highest power of the variable. 4x⁵ − x + 2 has degree 5, and the fundamental theorem of algebra says it has exactly 5 complex roots, counted with multiplicity.

How do I find the roots of a polynomial?

Factor if you can, use the rational root theorem to test candidates, or solve numerically. Each root r gives a factor (x − r).

What is the remainder theorem?

Dividing p(x) by x − r leaves remainder p(r). If p(r) = 0, then x − r is a factor.

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